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A050155
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Triangle T(n,k), k>=0 and n>=1, read by rows defined by: T(n,k) = (2k+3)*binomial(2n,n-k-1)/(n+k+2).
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4
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1, 3, 1, 9, 5, 1, 28, 20, 7, 1, 90, 75, 35, 9, 1, 297, 275, 154, 54, 11, 1, 1001, 1001, 637, 273, 77, 13, 1, 3432, 3640, 2548, 1260, 440, 104, 15, 1, 11934, 13260, 9996, 5508, 2244, 663, 135, 17, 1, 41990, 48450, 38760, 23256, 10659, 3705, 950, 170, 19, 1
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OFFSET
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1,2
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COMMENTS
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T(n-2k-1,k) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 2k+2 (cf. Zoran Sunic reference) . - Benoit Cloitre, Oct 07 2003
Number of lattice paths from (0,0) to (n,n) with steps E=(1,0) and N=(0,1) which touch but do not cross the line x-y=k+1 . - Herbert Kociemba, May 24 2004
Number of standard tableaux of shape (n+k+1, n-k-1). - Emeric Deutsch, May 30 2004
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LINKS
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FORMULA
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Sum_{ k = 0, .., n-1} T(n, k) = binomial(2n, n-1) = A001791(n).
For the column k : expansion of x^(k+1)C^(2k+3) where C = (1-(1-4*x)^(1/2)/(2*x) is the g.f. of Catalan numbers A000108. - Philippe Deléham, Feb 03 2004
T(n, k)=(2k+3)binomial(2n+2, n+k+2)/(n+k+3)=C(2n+2, n+k+2)-C(2n+2, n+k+3) [offset (0, 0)]. - Paul Barry, Jul 06 2005
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EXAMPLE
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1;
3, 1;
9, 5, 1;
28, 20, 7, 1;
90, 75, 35, 9, 1;
297, 275, 154, 54, 11, 1;
. . .
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MAPLE
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T:= (n, k)-> (2*k+3)*binomial(2*n, n-k-1)/(n+k+2):
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MATHEMATICA
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T[n_, k_] := (2*k + 3)*Binomial[2*n, n - k - 1]/(n + k + 2);
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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